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Results tagged with homotopy-theory
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user 139854
Homotopy theory is an important sub-field of algebraic topology. It is mainly concerned with the properties and structures of spaces which are invariant under homotopy. Chief among these are the homotopy groups of spaces, specifically those of spheres. Homotopy theory includes a broad set of ideas and techniques, such as cohomology theories, spectra and stable homotopy theory, model categories, spectral sequences, and classifying spaces.
6
votes
1
answer
248
views
Universal model category as a $\text{sSet}$-enriched co-completion
It's a standard fact that given a small category $\mathcal{C},$ the category of pre-sheaves $\text{Psh}(\mathcal{C})$ is the free co-completion of it.
I'm sure this can be done not only for $\text{Se …
- 1,514
4
votes
1
answer
345
views
Contractibility of the category of cosimplicial resolutions
Let $\gamma : \mathcal{C} \to \mathcal{M}$ be a functor and define a cosimplicial resoultion of $\gamma$ as a functor $\Gamma: \mathcal{C} \to \mathcal{M}^{\Delta}$ such that
$\Gamma C$ is Reedy cofi …
- 1,514
4
votes
1
answer
280
views
When are filtered colimits of (trivial) cofibrations still (trivial) cofibrations?
Let $\mathcal{M}$ be a
locally finitely presentable model category, cofibrantly generated by
two sets $\mathcal{I}$ and $\mathcal{J}$ of cofibrations and trivial
cofibrations with presentable domain …
- 1,514
4
votes
0
answers
173
views
Is the Reedy model structure cofibrantly generated?
I am reading about the Reedy model structure from Hovey's book and
I was wondering if the Reedy model structure on $\mathcal{M}^{\Delta}$
is cofibrantly generated by a small set of arrows and permits …
- 1,514
4
votes
2
answers
317
views
Quillen pairs / $\infty$-adjunctions / adjunctions of homotopy categories
Some of the examples of $\infty$-categories are those arising from model categories. I would like to ask: what is the relationship between Quillen adjunctions between model categories and adjoint pair …
- 1,514
14
votes
2
answers
843
views
sSet-enriched categories, quasi-categories and the model-independent theory
sSet-enriched categories are one possible model for $(\infty,1)$-categories, by the work of Bergner and others. They are probably the most important model from the point of view of getting actual exam …
- 1,514
21
votes
2
answers
2k
views
Mark Hovey's open problems in the theory of model categories
Mark Hovey maintains a list of open problems in model category theory. I think this list is quite old, and I don't know if Hovey is still updating it or not.
My question is:
i) which of the 13 prob …
- 1,514
26
votes
1
answer
1k
views
What's motivic about $\mathbb{A}^1$-homotopy theory? What's motivic about correspondences?
I come today to mathoverflow to showcase some genuine confusion about the motivic world. I want to ask some questions before actually starting to study the subject, to build some sense of direction.
I …
- 1,514